   Chapter 17.2, Problem 20E

Chapter
Section
Textbook Problem

Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters.20. y" – 2y' – 3y = x + 2

(a)

To determine

To solve: The differential equation by using method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

y2y3y=x+2 (1)

Consider the auxiliary equation.

r22r3=0 (2)

Roots of equation (2) are,

r=(2)±(2)24(1)(3)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=2±42=3and1

Write the expression for the complementary solution of two real roots.

yc(x)=c1er1x+c2er2x (3)

Substitute 3 for r1 and 1 for r2 in equation (3),

yc(x)=c1e3x+c2e1x

yc(x)=c1e3x+c2ex (4)

Consider G(x)=x+2 is a polynomial degree of 1.

If Right hand side (RHS) of a differential equation contains 1st order polynomial. The trail solution yp(x) for this case can be expressed as follows.

y(x)=(Axn+Bxn1+Z) (5)

Here,

n is the order of a polynomial, and

A,B,Z, are the constants.

Write trail solution for equation (1) using equation (5).

yp(x)=Ax1+B11

yp(x)=Ax+B (6)

Differentiate equation (6) with respect to x

(b)

To determine

To solve: The differential equation by using method of variation of parameters.

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